7th and 8th grade contest questions, 8th grade math contest questions

1 All Answers: A straight line must pass through b, and if there is no negative value, the triangle is an equilateral triangle. If there are negative values, you can't form a triangle.

Sort out (a+b) c=(b+c) a=(c+a) b=p to obtain,a+b cp,b+c ap,c+a bp,add the three formulas to get 2(a+b+c)=(a+b+c)p,if a+b+c is not equal to 0,just p 2,straight line through 1,2,3 quadrants,a=b=c,if a+b+c=0,p -1, straight line through 2,3,4 quadrants,a,b, c must have a negative value, and it cannot form a triangle.

From (b+c) a=(c+a) b and abc≠0 can obtain b 2-a 2=(b+a)(b-a)=c(a-b), and b+a=-c, i.e., p=-1, so y=-x-1, must pass.

2. Four quadrants. Because a, b, and c are not zero, and a+b+c=0, there must be a negative value in a, b, and c, and it is not possible to form a triangle from the perspective of the given title.

No matter how many hours it is, assuming that it is x hours to arrive together, the car transports a group of 40 people first, and the other 80 people are walking, and when they are almost at the end, they are put down, let them go, and then go to pick up the other 40 people, and after receiving it, they will soon go to the end point to pick up the other 40 people, and then send them back to the end point, at this time the three groups of people arrive at the same time. Then there are always two groups of people walking, and the car is always walking, so it is equivalent to walking back and forth on this road 5 times.

10*2*x+80*x=91*5

x = hours. It can be reached in hours at the earliest.

It took 5 minutes for the boat to catch up with the map from U-turn, indicating that when the boat caught up with the map, the boat and the map were in 5 minutes, and the boat traveled more than the map, and the ship still water speed was 5

This distance is the distance between the map falling into the water and the boat falling off the bridge, and at this time the boat is going against the water, the map is drifting, and their speed is still water speed, so this person finds that the map falls into the water after (b) 5 minutes.

Solution: This question is equivalent to ,...... 1, 2, 3In 2004, k-1 numbers were selected, so that no three numbers could be the length of the three sides of a triangle with unequal sides, and what was the maximum value of k that met this condition.

For arrays that meet the above conditions, when k = 4, the three smallest numbers are 1, 2, 3From this, the array can be continuously expanded, as long as the number added is greater than or equal to the sum of the two largest values in the obtained array, so that in order to reach the maximum k, the number of optional additions is equal to the sum of the two largest numbers in the obtained array, so that we get:

There are 16 numbers in total, for any set of arrays that meet the above conditions, a1, a2, ....an, apparently there is always ai greater than or equal to the ith number in , so n is 16 k-1, so that the minimum value of k is 17

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